Entanglement and Bond Fluctuations in Random Singlet Phases

The set of valence-bond states --- states in which localized spin-1/2 particles are correlated in singlet pairs said to be connected by valence bonds --- provides a useful basis for visualizing singlet ground states of quantum spin systems. For example, the ground state of the uniform one-dimensional nearest-neighbor spin-1/2 antiferromagnetic (AFM) Heisenberg model (the prototypical spin-liquid state) can be viewed as a strongly fluctuating liquid of valence bonds with a power-law length distribution. This intuitive picture directly reflects the long-range spin correlations in this state, as well as the existence of gapless excitations created by breaking long bonds. Valence-bond states also play a key role in describing the physics of random spin-1/2 AFM Heisenberg chains. For these systems, it was shown by Fisher, using a real space renormalization group analysis, that on long-length scales the ground state is described by a single valence-bond state known as a random singlet state. This single valence-bond state should be viewed as a caricature of the true ground state, which will certainly exhibit bond fluctuations on short-length scales. In valence-bond Monte Carlo (VBMC) simulations valence-bond states are used to stochastically sample singlet ground states of quantum spin systems. One of the appealing features of VBMC is that if one imagines viewing the sampled valence-bond states over many Monte Carlo time steps the resulting ''movie" would correspond closely to the intuitive resonating valence bond picture described above. For random Heisenberg chains (and related models) VBMC should therefore provide a useful method for directly studying the phenomenon of random singlet formation on long-length scales, while at the same time capturing the short-range fluctuations which will always be present. In this dissertation I present results of VBMC studies for a class of models which include the uniform and random spin-1/2 AFM Heisenberg chains, as well as models describing chains of interacting non-Abelian quasiparticles --- exotic quasiparticles conjectured to exist in certain fractional quantum Hall states. In addition to numerically computing and analyzing the so-called valence-bond entanglement scaling in these models, I introduce a new quantity which I refer to as the valence-bond fluctuation (the central new result and the main contribution of this dissertation). It is shown that this quantity, which is easy to compute in valence-bond Monte Carlo, provides a direct signature of random singlet phase formation by essentially allowing one to directly ''see" the ''locking" of the ground state into a particular valence-bond state on long-length scales. A detailed scaling analysis of this new quantity is then used to extract the dependence of the fluctuation length scale on disorder strength. Where possible, the results are compared to previous numerical and analytic work on the relevant models.